Expert: Scott A Wilson - 10/10/2011

Question
hello
i would like to know what the logic behind the square of a symmetric matrix being symmetric is? what i mean to say is if A is a symmetric matrix, then how do we show (in other words, what is the logic) that A^2 is also symmetric without assuming numbers or variables for matrix A?
looking forward to a prompt reply.
Thank you
Saman

Answer
When two matrices are multiplied toegether, the answere of the element in the ith row and jth column is the result of multiplying the ith row of the 1st matrix by the jth column of the 2nd matrix.

When a reflective matrix is multiplied by itself, the ith row is the same as the jth column and the jth row is the same as the ith column.  From this, the element gotten in position (i,j) is the same as the element gotten in (j,i).  This makes the resulting matrix symmetric as well.

Suppose we had the matrix
1   2   3   4
2   1   6   5
3   6   4   7
4   5   7   3

To get the element in row 1, column 3, we would multiply the 1st row by the 3rd column.
The same result would be gotten in row 3, column 1, for the 1st row and 1st column are the same and the 3rd row and 3rd column are the same.

The resulting matrix would be
30   42   55   47
42   66   71   70
55   71   110   91
47   70   91   99

As can be seen, this is still reflective.

To be more specific, the 91 in is gotten by multiplying the 4th row by the 3rd column or the 4th column by the 3rd row.  That is, it is 3*4 + 6*5 + 4*7 + 7*3 = 12 + 30 + 28 + 21 = 91 and
4*3 + 5*6 + 7*4 + 3*7 = 12 + 30 + 28 + 21 = 91.